HIGH-FREQUENCY RESOLVENT ESTIMATES ON ASYMPTOTICALLY EUCLIDEAN WARPED PRODUCTS

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1 HIGH-FREQUENCY RESOLVENT ESTIMATES ON ASYMPTOTICALLY EUCLIDEAN WARPED PRODUCTS HANS CHRISTIANSON Abstract. We consider the resolvent on asymptotically Euclidean warped product manifolds in an appropriate -Gevrey class, with trapped sets consisting of only finitely many components. We prove that the high-frequency resolvent is either bounded by C ǫ λ ǫ for any ǫ >, or blows up faster than any polynomial (at least along a subsequence). A stronger result holds if the manifold is analytic. The method of proof is to exploit the warped product structure to separate variables, obtaining a one-dimensional semiclassical Schrödinger operator. We then classify the microlocal resolvent behaviour associated to every possible type of critical value of the potential, and translate this into the associated resolvent estimates. Weakly stable trapping admits highly concentrated quasimodes and fast growth of the resolvent. Conversely, using a delicate inhomogeneous blowup procedure loosely based on the classical positive commutator argument, we show that any weakly unstable trapping forces at least some spreading of quasimodes. As a first application, we conclude that either there is a resonance free region of size Imλ C ǫ Reλ 1 ǫ for any ǫ >, or there is a sequence of resonances converging to the real axis faster than any polynomial. Again, a stronger result holds if the manifold is analytic. As a second application, we prove a spreading result for weak quasimodes in partially rectangular billiards. 1. Introduction In this paper, we consider manifolds which have a warped product structure and are asymptotically Euclidean with a certain -Gevrey regularity. Our main result is that the cutoff resolvent is either (almost) bounded or blows up faster than any polynomial. Of course, the proof gives much more information than this simple statement, but for aesthetic reasons we prefer to phrase it in this fashion. Let us state the main result. Theorem 1. Let X be a -Gevrey smooth G τ, τ <, warped product manifold without boundary which is a short range perturbation of Euclidean space (with one or two infinite ends). Assume also that the trapped set on X has finitely many connected components. Let be the Laplace-Beltrami operator on X. Then either 1: For every ϕ C c (X) and every ǫ >, there exists C ǫ > such that (1.1) ϕ( (λ i) 2 ) 1 ϕ C ǫ λ ǫ, λ 1. or 2: For every N >, there exists ϕ Cc (X), C N >, and a sequence λ j R, λ j, such that (1.2) ϕ( (λ j i) 2 ) 1 ϕ C N λ j N. 21 Mathematics Subject Classification. 35B34,35S5,58J5,47A1. 1

2 2 H. CHRISTIANSON Remark 1.1. The warped product structure at infinity can be replaced by a number of different non-trapping infinite ends, using the recent gluing theorem of Datchev-Vasy [DV12] (see also [Chr8] and Appendix A). The dichotomy in Theorem 1 is from the following idea: if there is any weakly stable trapping on X, then there are well-localized quasimodes, and we are in Case 2 of the Theorem. If all the trapping is at least weakly unstable, we need to prove there is weak microlocal non-concentration near each connected component of the trapped set, as well as prove that there is no strong tunneling between different connected components (i.e. that the different connected components at the same energy don t talk to each other too much). As we shall see, the worst behaviour in (1.1) in Theorem 1 comes from weakly unstable trapping which is infinitely degenerate. Since such trapping cannot occur on an analytic manifold, there is a nice improvement in this case, given in the next Corollary. Corollary 1.2. In addition to the assumptions of Theorem 1, assume that the manifold X is analytic. Then either 1: There exists δ > such that for every ϕ C c (X), there is a constant C > for which we have the estimate (1.3) ϕ( (λ i) 2 ) 1 ϕ C λ δ, λ. or 2: For every N >, there exists ϕ Cc (X), C N >, and a sequence λ j R, λ j, such that (1.4) ϕ( (λ j i) 2 ) 1 ϕ C N λ j N. Remark 1.3. Upon rescaling to a semiclassical problem, Theorem 1 states that the semiclassical cutoff resolvent is either controlled by h 2 ǫ for any ǫ >, or blows up faster than h N for any N. The corollary states that if the manifold is analytic, the first possibility can be replaced with h 2+δ for some δ > fixed, depending on the trapping. As usual, high energy resolvent estimates imply there are regions free of resonances by simple perturbation of the spectral parameter. On the other hand, the proof of the alternative large growth of the resolvent along a subsequence proceeds by quasimode construction. Then if our metric has a complex analytic extension outside of a compact set, we can apply the results of Tang-Zworski [TZ98] to conclude existence of resonances. This results in the following Corollary. Corollary 1.4. In addition to the assumptions of Theorem 1, assume X admits a complex analytic extension outside of a compact set so that resonances may be defined by complex scaling. Then either 1: For every ǫ > there is a constant C ǫ > such that the region {λ C : Imλ C ǫ Reλ 1 ǫ, λ 1} is free of resonances and the estimate (1.1) holds there (with a suitably modified constant), or 2: For every N >, there exists a sequence of resonances {λ j } such that Imλ j Reλ j N, λ j.

3 RESOLVENT ESTIMATES 3 In particular, if X is analytic, then either there exists δ > and C > such that the region {λ C : Imλ C Reλ 1+δ, λ 1} is free from resonances, or there is a sequence converging to the real axis at an arbitrarily fast polynomial rate Resolvents and the local smoothing effect. One of the many motivations for studying resolvents and resolvent estimates is to understand the local smoothing effect for the Schrödinger equation on manifolds with trapping. It is well known(see, for example, [Tao6, Doi96]) that on asymptotically Euclidean manifolds without trapping, solutions to the Schrödinger equation enjoy a 1/2 derivative local smoothing effect. This says that, locally in space, and on average in time, solutions are 1/2 derivative smoother than the initial data. To be precise, let X be such a manifold, the Laplacian on X, u a Schwartz function on X, and χ C c (X) a cutoff function. Then the following estimate holds true for any T > : T χe it u 2 H 1/2 (X) dt C T u 2 L 2 (X). There are several ways to prove such an estimate; one way proceeds through resolvent estimates (see Section 5 below). A nice benefit of using the resolvent formalism to understand local in time local smoothing (that is, for finite T) is that one really sees how the spectral estimates are related to the smoothing effect. Since one only needs a resolvent estimate in a fixed strip near the real axis, if one is in a situation where the limiting resolvent blows up, one simply uses the trivial bound away from the real axis to get a zero derivative smoothing effect (or just integrates the L 2 (X) mass in time). However, if the limiting resolvent has some decay, then there is a non-trivial local smoothing estimate. This is the case, for example, if the manifold is analytic and all of the trapping is at least weakly unstable. Let us state this as a corollary: Corollary 1.5. Let X be an analytic warped product manifold so that all of the assumptions of Corollary 1.2 hold. Assume also that every connected component of the trapped set is at least weakly unstable, so that conclusion 1 of Corollary 1.2 holds for some δ >. Then for all χ C c (X), u S(X), and T >, there exists C T > such that T χe it u 2 H δ/2 (X) dt C T u 2 L 2 (X). Acknowledgements. This research was partially supported by NSF grant DMS The author would like to thank K. Datchev, L. Hillairet, J. Metcalfe, E. Schenck, M. Taylor, A. Vasy, and J. Wunsch for many helpful and stimulating discussions. 2. Preliminaries 2.1. The geometry. We have assumed that our manifold X has a warped product structure with one or two infinite ends which are short range perturbations of R n. This means we are considering the manifold X = R x Ω n 1 θ (or X = R + x Ω n 1 θ if one infinite end), equipped with the metric g = dx 2 +A 2 (x)g θ,

4 4 H. CHRISTIANSON where A C is a smooth function, A ǫ > for some epsilon (or A(x) = x for x > near if one infinite end), and G θ is the metric on a smooth compact n 1 dimensional Riemannian manifold Ω n 1 without boundary. The short range assumption means that as x, we have α (g g E ) C α x 2 α, where g E = dx 2 +x 2 G θ. This means that X is asymptotically Euclidean as x. This assumption merely allows us to use standard techniques to glue resolvent estimates together without worrying about trapping at infinity. The assumptions can of course be weakened to long-range perturbation (following Vasy-Zworski [VZ]), but this paper is really about the local phenomenon of trapping rather than having the most general infinity. Weusethenotationθ Ω n 1 todenotethe angular directions. This is in analogue with the case of spherically symmetric warped product spaces where Ω n 1 = S n 1 is the sphere and X is asymptotically R n. From this metric, we get the volume form dvol = A(x) n 1 dxdσ, where σ is the volume measure on Ω n 1. The Laplace-Beltrami operator acting on -forms is computed: f = ( 2 x +A 2 Ω n 1 +(n 1)A 1 A x )f, where Ω n 1 is the (non-positive) Laplace-Beltrami operator on Ω n 1. We want to exploit the warped product structure to reduce spectral questions to a one-dimensional problem. Let us first conjugate to a problem on the flat cylinder. That is, let Tu(x,θ) = A (n 1)/2 (x)u(x,θ) so that = T T 1 is essentially selfadjoint on L 2 (dxdσ), where σ is the usual volume measure on Ω n 1. We have where = 2 x A 2 (x) Ω +V 1 (x), V 1 (x) = n 1 A A 1 (n 1)(n 3) (A ) 2 A Separating variables we write for u L 2 (dxdσ) u(x,θ) = l,k u lk (x)ϕ lk (θ), where ϕ lk (θ) are the eigenfunctions on Ω n 1 with eigenvalue λ 2 k. Then u = l,k ϕ lk (θ)q k u lk, where Q k ϕ(x) = ( x 2 +λ 2 ka 2 (x)+v 1 (x))ϕ(x). Setting h = λ 1 k and rescaling, we end up with the semiclassical operator P(h)ϕ(x) = ( h 2 2 x +V(x))ϕ(x), where V(x) = A 2 (x)+h 2 V 1 (x). We sometimes will write V (x) = A 2 (x) for the principal part of the effective potential.

5 RESOLVENT ESTIMATES 5 The semiclassical versions of Theorem 1 and Corollary 1.2 are given in the following. Theorem 2. Under the assumptions above, either 1: For every ϕ C c (R) and every ǫ > there exists C ǫ > such that ϕ( h 2 2 x +V (z i)) 1 ϕ C ǫ h 2 ǫ, z I, for a compact interval I, or 2: For every N >, there exists ϕ C c (X), C N >, and z R, z, such that ϕ( h 2 2 x +V (z i)) 1 ϕ C N h N, along a subsequence as h +. Corollary 2.1. In addition to the assumptions of Theorem 2, assume that the manifold X is analytic. 1: There exists δ > such that, for every ϕ C c (R) there is a constant C > for which we have the following estimate ϕ( h 2 2 x +V (z i)) 1 ϕ Ch 2+δ, z I, for a compact interval I, or 2: For any N >, there exists ϕ C c (X), C N >, and z R, z, such that ϕ( h 2 2 x +V (z i)) 1 ϕ C N h N along a subsequence as h The -Gevrey class G τ. Our manifolds already have very nice geometry as x, and moreover we have separated variables. Since Ω n 1 is a C compact manifold without boundary, the only additional regularity assumptions we need to impose will be at the critical elements of the manifold X, that is, at the critical points of the function A(x). In order to have a meaningful symbol class (especially once we are working with the calculus of 2 parameters), we need to know that near the critical elements, the function A is not too far away from being analytic. For this, we introduce the following -Gevrey classes of functions with respect to order of vanishing. For τ <, let G τ(r) be the set of all smooth functions f : R R such that, for each x R, there exists a neighbourhood U x and a constant C such that, for all s k, k xf(x) k xf(x ) C(k!) C x x τ(k s) s xf(x) s xf(x ), x x in U. This definition says that the order of vanishing of derivatives of a function is only polynomially worse than that of lower derivatives. Every analytic function is in one of the -Gevrey classes Gτ for some τ <, but many more functions are as well. For example, the function { exp( 1/x p ), for x >, f(x) =, for x is in G p+1, but f(x) = { exp( exp(1/x)), for x >,, for x

6 6 H. CHRISTIANSON is not in any -Gevrey class for finite τ. This implies that the -Gevrey class contains a rich subset of functions with compact support as well as functions which are constant on intervals. The -Gevrey class assumption will only come in to play in the case of infinitely degenerate critical points (see Subsection 3.4) Semiclassical calculus with 2 parameters. Following Sjöstrand-Zworski [SZ7, 3.3] and [CW11], we introduce a calculus with two parameters. We will not present the proofs in the following lemmas, as they have appeared in several other places, but merely include the statements for the reader s convenience, as well as pointers to where proofs can be found. For α [,1] and β 1 α, we let S k,m, m α,β (T (R n )) := { = a C ( R n (R n ) (,1] 2) : x ρ γ ξ a(x,ξ;h, h) C ργ h m h m ) α ρ +β γ } ( h ξ k γ. h Throughout this work we will always assume h h. We let Ψ k,m, m α,β denote the corresponding spaces of semiclassical pseudodifferential operators obtained by Weyl quantization of these symbols. We will sometimes add a subscript of h or h to indicate which parameter is used in the quantization; in the absence of such a parameter, the quantization is assumed to be in h. The class S α,β (with no superscripts) will denote S,, α,β for brevity. In [SZ7] (for the homogeneous case α = β = 1/2), and in [CW11] (for the inhomogeneous case α β), it is observed that the composition in the calculus can be computed in terms of a symbol product that converges in the sense that terms improve in h and ξ orders, but not in h orders. This happens because when α+β = 1, the (h α,h β ) calculus is marginal, which is what the rescaling (blowup) and introduction of the second parameter h accomplishes. In the sequel, we will always assume we are in the inhomogeneous marginal case: α+β = 1. If α + β < 1, then of course the calculus is no longer marginal and computations become much easier. By the same arguments employed in [SZ7] (see [CW11]), we may easily verify that the calculus Ψ α,β is closed under composition: if a S k,m, m α,β and b S k,m, m α,β then Op w h(a) Op w h(b) = Op w h(c) with c S k+k,m+m, m+ m α,β. In addition, as in [CW11], we have a symbolic expansion for c in powers of h. We have the following Lemma from [CW11], which is a more general version of [SZ7, Lemma 3.6]: Lemma 2.2. Suppose that a,b S α,β, and that c w = a w b w. Then N ( ) k 1 ih (2.1) c(x,ξ) = k! 2 σ(d x,d ξ ;D y,d η ) a(x,ξ)b(y,η) x=y,ξ=η +e N (x,ξ), k=

7 RESOLVENT ESTIMATES 7 where for some M (2.2) γ e N C N h N+1 γ 1+γ 2=γ sup (x,ξ) T R n (y,η) T R n where σ(d) = σ(d x,d ξ ;D y,d η ) as usual, and sup Γα,β,ρ,γ (D)(σ(D)) N+1 a(x,ξ)b(y,η), ρ M,ρ N 4n Γ α,β,ρ,γ (D) = (h α (x,y),h β (ξ,η) )) ρ γ1 γ2. As a particular consequence we notice that if a S α,β (T R n ) and b S(T R n ) then N 1 (2.3) c(x,ξ) = k! (ihσ(d x,d ξ ;D y,d η )) k a(x,ξ)b(y,η) x=y,ξ=η k= { +O Sα,β (h N+1 max ( h/h) (N+1)α,( h/h) (N+1)β}). We will let B denote the blowdown map (2.4) (x,ξ) = B(X,Ξ) = ((h/ h) α X,(h/ h) β Ξ). The spaces of operators Ψ h and Ψ h are related via a unitary rescaling in the following fashion. Let a S k,m, m α,β, and consider the rescaled symbol ( ) αx, ( ) ) βξ a( h/ h h/ h = a B S k,m, m,. Define the unitary operator T h, hu(x) = Op w h(a B)T h, hu = T h, hop w h(a)u. ( )nα (( ) 2 αx ) h/ h u h/ h, so that 3. The trapping In order to prove Theorem 2 and Corollary 2.1, we consider the critical points of the potential V(x), or more specifically the critical points of the principal part, V (x) = A 2, of V = A 2 + h 2 V 1. The assumption that the trapped set has only finitely many connected components implies that the potential V (x) has only finitely many critical values. We break the analysis of the critical values into those for which the Hamiltonian flow of the principal part of our symbol, p = ξ 2 +V (x), is locally unstable (either genuinely unstable or of transmission inflection type), and those for which the Hamiltonian flow is stable. This leads to the dichotomy in Theorem 2 and Corollary 2.1. The idea is that, if there is a critical value for which the Hamiltonian flow is stable, then we can immediately construct very good quasimodes and reach the second conclusions in Theorem 2 and Corollary 2.1. This is relatively straightforward and written in Subsection 3.6. On the other hand, if there is no stable trapping, then all trapping is unstable, consisting of disjoint critical sets, and even if two critical sets exist at the same potential energy level, they must be separated by an unstable maximum critical value at a higher potential energy level (otherwise there would be a minimum in between, and hence at least weakly stable trapping), so they do not see each other. That is to say, the weakly stable/unstable manifolds of the separating maximum form a separatrix in the reduced phase space. This allows us to glue together microlocal estimates near each critical set, and the resolvent estimate is then simply the worst

8 8 H. CHRISTIANSON of these estimates. Hence it suffices to classify microlocal resolvent estimates in a neighbourhood of any of these unstable critical sets. This is accomplished in Subsections In this sense, this section contains a catalogue of microlocal resolvent estimates. It is important to note at this point that for unstable trapping of finite degeneracy, the relevant resolvent estimates are all o(h 2 ), that is to say, the sub-potential h 2 V 1 is always of lower order. If the trapping is unstable but infinitely degenerate, we need to work harder to absorb the sub-potential. The -Gevrey assumption will be important here Unstable nondegenerate trapping. Unstable nondegenerate trapping occurs when the potential V has a nondegenerate maximum. As mentioned previously, let us for the time being consider the operator Q = h 2 x +V (x) z, where V (x) = A 2 (x). To say that x = is a nondegenerate maximum means that x = is a critical point of V (x) satisfying V () =, V () <, and then the Hamiltonian flow of q = ξ 2 +V (x) near (,) is { ẋ = 2ξ, ξ = V (x) x, so that the stable/unstable manifolds for the flow are transversal at the critical point (, ). The following result as stated can be read off from [Chr7,Chr1,Chr11], and has also been studied in slightly different contexts in [CdVP94a, CdVP94b] and [BZ4], amongst many others. We only pause briefly to remark that, since the lower bound on the operator Q is of the order h/log(1/h) h 2, the same result applies equally well to Q+h 2 V 1. Proposition 3.1. Suppose x = is a nondegenerate local maximum of the potential V, V () = 1. For ǫ > sufficiently small, let ϕ S(T R) have compact support in { (x,ξ) ǫ}. Then there exists C ǫ > such that (3.1) Qϕ w u C ǫ h log(1/h) ϕw u, z [1 ǫ,1+ǫ] Unstable finitely degenerate trapping. In this subsection, we consider an isolated critical point leading to unstable but finitely degenerate trapping. That is, we now assume that x = is a degenerate maximum for the function V (x) = A 2 (x) of order m 2. If we again assume V () = 1, then this means that near x =, V (x) 1 x 2m. Critical points of this form were studied in [CW11], but the proof can also be more or less deduced from the proofs of Propositions 3.6 and 3.8 below. We only remark briefly that again, since the lower bound on the operator Q is of the order h 2m/(m+1) h 2, the estimate applies equally well to Q+h 2 V 1. Proposition 3.2. Let Q = h 2 2 x + V (x) z. For ǫ > sufficiently small, let ϕ S(T R) have compact support in { (x,ξ) ǫ}. Then there exists C ǫ > such that (3.2) Qϕ w u C ǫ h 2m/(m+1) ϕ w u, z [1 ǫ,1+ǫ]. Remark 3.3. In [CW11], it is also shown that this estimate is sharp in the sense that the exponent 2m/(m+1) cannot be improved.

9 RESOLVENT ESTIMATES Finitely degenerate inflection transmission trapping. We next study the case when the potential has an inflection point of finitely degenerate type. That is, let us assume the point x = 1 is a finitely degenerate inflection point, so that locally near x = 1, the potential V (x) = A 2 (x) takes the form V (x) C 1 1 c 2 (x 1) 2m2+1, m 2 1 where C 1 > 1 and c 2 >. Of course the constants are arbitrary (chosen to agree with those in [CM13]), and c 2 could be negative without changing much of the analysis. This Proposition and the proof are in [CM13], and as we will once again revisit the proof of this Proposition in Subsection 3.5, we will omit it at this point. But one last time, let us observe that since the lower bound on the operator Q is of the order h (4m2+2)/(2m2+3) h 2, the estimate applies equally well to the operator Q+h 2 V 1. Proposition 3.4. For ǫ > sufficiently small, let ϕ S(T R) have compact support in { (x 1,ξ) ǫ}. Then there exists C ǫ > such that (3.3) Qϕ w u C ǫ h (4m2+2)/(2m2+3) ϕ w u, z [C 1 1 ǫ,c 1 1 +ǫ]. Remark 3.5. We remark that in this case, [CM13] shows once again that this estimate is sharp in the sense that the exponent (4m 2 + 2)/(2m 2 + 3) cannot be improved Unstable infinitely degenerate and cylindrical trapping. In this subsection, westudythecasewheretheprincipalpartofthepotentialv(x) = A 2 (x)+ h 2 V 1 (x) has an infinitely degenerate maximum, say, at the point x =. Let V (x) = A 2 (x). As usual, we again assume that V () = 1, so that V (x) = 1 O(x ) in a neighbourhood of x =. Of course this is not very precise, as V could be constant in a neighbourhood of x = and still satisfy this, and the proof must be modified to suit these two cases. So let us first assume that V () = 1, and V (x) vanishes to infinite order at x =, however, ±V (x) < for ±x >. That is, the critical point at x = is infinitely degenerate but isolated. Our microlocal spectral theory result is then that the microlocal cutoff resolvent is bounded by O η (h 2 η ) for any η >. In order to state the result, let Q = h 2 2 x +V(x) z = h 2 2 x +A 2 (x)+h 2 V 1 (x) z. Proposition 3.6. For ǫ > sufficiently small, let ϕ S(T R) have compact support in { (x,ξ) ǫ}. Then for any η >, there exists C ǫ,η > such that (3.4) Qϕ w u C ǫ,η h 2+η ϕ w u, z [1 ǫ,1+ǫ]. Remark 3.7. As this is the limiting case as m of Proposition 3.2, we believe the optimal lower bound in this case is h 2 /γ(h) for some γ(h). This is further suggested by a microlocal scaling heuristic. However, various attempts to tighten up the argument to get the better lower bound seem to fail. It would be very interesting to determine if a lower bound of h 2 /γ(h) or even h 2 holds. For our next result, we consider the case where there is a whole cylinder of unstable trapping. That is, we assume the principal part of the effective potential V (x) has a maximum V (x) 1 on an interval, say x [ a,a], and that ±V (x) <

10 1 H. CHRISTIANSON for ±x > a. Our main result in this case says that the microlocal cutoff resolvent is again controlled by h 2 η for any η >. Let us again set Q = h 2 2 x +V(x) z. Proposition 3.8. For ǫ > sufficiently small, let ϕ S(T R) have compact support in { x a + ǫ, ξ ǫ}. Then for any η >, there exists C ǫ,η > such that (3.5) Qϕ w u C ǫ,η h 2+η ϕ w u, z [1 ǫ,1+ǫ]. Remark 3.9. For similar reasons, we expect the optimal lower bound in this case should be h 2. Proof. The proof of these Propositions is very similar, so we put them together. We will first prove Proposition 3.6, and then point out how the proof must be modified to get Proposition 3.8. The idea of the proof of Proposition 3.6 (and indeed Proposition 3.8) is to add a small h-dependent bump with a finitely degenerate maximum, and then use the result of Proposition 3.2. Of course the bump has to be sufficiently small that the operator Q is close to the perturbed operator. Choose a point x = x (h) > and ǫ > so that x is the smallest point such that xv (x) h (h), x x ǫ, where (h) will be determined later. As long as (h) h, this implies that x = o(1). We remark that, of course, x depends also on the choice of (h), but for any, there is such a choice, since V vanishes to infinite order at x =. Further, as V (x) = O(x ) near x =, we have x h δ for any δ >. Fix m 2 to be determined later in the proof (m will be large), and choose also an even function f Cc ([ 2,2]) Gτ for some τ <, with f(x) = 1 1 2m x2m for x 1, and f (x) or x [,2]. For another parameter Γ(h) > to be determined, let and let and W h (x) = Γ(h)f(x/x ), V,h (x) = V (x)+w h (x) V h (x) = V(x)+W h (x) (see Figure 1). The parameter Γ(h) will be seen to be h 2+η for η >, η = O(m 1 ) as m. By construction, V(x) V h (x) W h Γ(h). Let Q 1 = (hd) 2 + V h with symbol q 1 = ξ 2 + V h. The Hamilton vector field H associated to the symbol q 1 is given by H = 2ξ x V h ξ ( ) Γ(h) = 2ξ x f (x/x )+V x (x)+h 2 V 1(x) ξ.

11 RESOLVENT ESTIMATES 11 V,h = 1+Γ(h) V = 1 x = x = x x = 2x Figure 1. The potential V and the modified potential V,h (in dashed). We will use the same change of coordinates and the same singular commutant as in [CW11], but we also have to track the loss coming from the coefficient Γ(h). For α = 1/(m+1), let Ξ = ξ (h/ h) mα, X = x (h/ h) α, so that in the new blown-up coordinates Ξ,X, ( (3.6) H = (h/ h) m 1 m+1 Ξ X (h/ h) (1 2m)/(m+1) V h((h/ h) α ) X) Ξ Let Λ(s) be defined as in [CW11] by fixing ǫ > and setting Λ(s) = s z 1 ǫ dz, so that Λ is a zero order symbol satisfying Λ(s) s for s near. Following [Chr7, Chr11, CW11], we define a(x,ξ;h) = Λ(Ξ)Λ(X)χ(x)χ(ξ) = Λ(ξ/(h/ h) mα )Λ(x/(h/ h) α )χ(x)χ(ξ), where χ(s) is a cutoff function equal to 1 for s < δ 1 and for s > 2δ 1 (δ 1 will be chosen shortly). Then a is bounded, and a symbol in X,Ξ : X α β Ξ a Cα,β. (Recall that x = (h/ h) α X and ξ = (h/ h) mα Ξ.) Using (3.6), it is simple to compute (3.7) with H(a) =(h/ h) m 1 m+1 χ(x)χ(ξ) ( Λ(Ξ) X 1 ǫ Ξ (h/ h) (1 2m)/(m+1) Λ(X)V h((h/ h) α X) Ξ 1 ǫ) +r (h/ h) m 1 m+1 g +r suppr { x > δ 1 } { ξ > δ 1 } (r comes from terms involving derivatives of χ(x)χ(ξ)). For X (h/ h) α x, we have with (h/ h) (1 2m)/(m+1) Λ(X)V h((h/ h) α X) Ξ 1 ǫ = Γ(h)x 2m Λ(X)X 2m 1 Ξ 1 ǫ +g 2, g 2 = (h/ h) (1 2m)/(m+1) Λ(X)(V +h 2 V 1)((h/ h) α X) Ξ 1 ǫ.

12 12 H. CHRISTIANSON Note that we always have Λ(x)V (x), so we expect the quantization of g 2 to be at least semibounded below. This is demonstrated in Lemma 3.11 below. For X (h/ h) α x and Ξ (h/ h) αm δ 1 consider g =χ(x)χ(ξ) ( Λ(Ξ) X 1 ǫ Ξ (h/ h) (1 2m)/(m+1) Λ(X)V h((h/ h) α X) Ξ 1 ǫ =Λ(Ξ)Ξ X 1 ǫ +Γ(h)x 2m Λ(X)X 2m 1 Ξ 1 ǫ +g 2 =λ 2( λ 1 Λ(Ξ)(λ 1 Ξ) X 1 ǫ +λ 2m 2 Γx 2m λλ(x)(λx) 2m 1 Ξ )+g 1 ǫ 2 =λ 2( Λ1 (Ξ )Ξ λ 1 X ) 1 ǫ +λ 2m 2 Γx 2m Λ 2 (X )(X ) 2m 1 λξ 1 ǫ +g 2 =:g 1 +g 2, where we have used the L 2 -unitary rescaling X = λx, Ξ = λ 1 Ξ, and λ > (small) will be determined in the course of the proof. The functions Λ j, j = 1,2, are defined by changing variables: and Λ 1 (Ξ ) = λ 1 Λ(Ξ) = λ 1 Λ(λΞ ), Λ 2 (X ) = λλ(x) = λλ(λ 1 X ). The error term g 2 is the term in the expansion of g coming from estimating using W h rather than V h. We will deal with g 2 in due course. We are now microlocalized on a set where X λ(h/ h) α x, Ξ λ 1 (h/ h) mα δ 1, and will be quantizing in the h-weyl calculus, so we need symbolic estimates on these sets. If λ 1 X δ 1, and λξ δ 1, and δ 1 > is sufficiently small, then Λ 1 (Ξ ) Ξ and Λ 2 (X ) X, so that g 1 is bounded below as follows: (3.8) g 1 min { λ 2,λ 2m Γx 2m } ((Ξ ) 2 +(X ) 2m ). Then the h-quantization of g 1 is bounded below microlocally on this set by this minimum times h 2m/(m+1) (see [CW11, Lemma A.2]). Now on the complementary set, we have one of either λ 1 X 1+ǫ or λξ is larger than, say, (δ 1 /2) 1+ǫ. We ( also need to keep track of the relative size of these two quantities. If λξ max λ 1 X ) 1+ǫ,(δ 1 /2) 1+ǫ then g 1 λ 2 Λ1 (Ξ )Ξ λ 1 X 1 ǫ (3.9) cλ 2 Λ1 (Ξ ) Ξ λξ = cλ Λ 1 (Ξ )sgn(ξ ) = cλ(λξ )sgn(ξ ) c δ1.

13 RESOLVENT ESTIMATES 13 Hencethe h-quantizationofg 1 isboundedbelowbyapositiveconstant,independent of h and h on this set. The remaining set is a bit more difficult. If λ 1 X 1+ǫ max ( λξ,(δ 1 /2) 1+ǫ), then (3.1) g 1 λ 2( Λ1 (Ξ )Ξ (h/ h) α x 1 ǫ +λ 2m 2 Γx 2m Λ 2 (X )(X ) 2m 1 λξ (1+ǫ)) = λ 2( λ 1 Λ(λΞ )Ξ (h/ h) α x 1 ǫ +λ 2m 2 Γx 2m λλ(λ 1 X )(X ) 2m 1 λξ (1+ǫ)) = λ 2( λ 1 Λ(λΞ )Ξ (h/ h) α x 1 ǫ +λ 2m Γx 2m Λ(λ 1 X )(X ) 2m 2 (λ 1 X ) λξ (1+ǫ)) c δ1 min{λ 2 (h/ h) α(1+ǫ) x 1 ǫ,λ 2m+2 Γx 2m } ((Ξ ) 2 +(X ) 2m 2 ), if λξ (δ 1 /2) 1+ǫ, c δ 1 (h/ h) α(1+ǫ) x 1 ǫ, if λξ (δ 1 /2) 1+ǫ. We now optimize the minimum in (3.1) to determine λ in terms of the other parameters: or Then the minimum is λ 2 (h/ h) α(1+ǫ) x 1 ǫ = λ 2m+2 Γx 2m, λ 2 = Γ 1/m (h/ h) α(1+ǫ)/m x 2+(1+ǫ)/m. λ 2 (h/ h) α(1+ǫ) x 1 ǫ = Γ 1/m (h/ h) α(1+ǫ)(m 1)/m x 3 ǫ+(1+ǫ)/m, and the h-quantization of g 1 on this set is bounded below microlocally by this number times h 2(m 1)/m (see [CW11, Lemma A.2]). Remark 3.1. We pause to remark that here is one place where alternative methods to optimize the lower bounds give worse results. For example, on the set where λξ (δ 1 /2) 1+ǫ λ 1 X 1+ǫ, we could estimate g 1 from below using only the second term. This gives a lower bound of c δ 1 Γx 2m, which is much worse than that computed above. Finally, recalling that eventually h > will be fixed and h h, taking the worst lower bound from (3.8) through (3.1), we obtain for a function u with h-wavefront set contained in the set where λ 1 X (h/ h) α x, λξ (h/ h) mα δ 1, Op h(g 1 )u,u Γ 1/m (h/ h) α(1+ǫ)(m 1)/m x 3 ǫ+(1+ǫ)/m h 2(m 1)/m u 2.

14 14 H. CHRISTIANSON On the other hand, if (h/ h) α x X (h/ h) α δ 1, we have (3.11) (h/ h) (1 2m)/(m+1) Λ(X)V h((h/ h) α X) Ξ 1 ǫ = (h/ h) (1 2m)/(m+1) sgn(x)b(x) (h/ h) α X (h/ h) α X V ((h/ h) α X) Ξ 1 ǫ +g 3 ( ) B(X) (h/ h) (1 2m)/(m+1) h (h/ h) α X (h) O(h2 ) Ξ 1 ǫ +g 3 C C h(2 m)/(m+1) h (2m 1)/(m+1) (h) h(2 m)/(m+1) h (2m 1)/(m+1) (h) (m 1)/(m+1) h3/(m+1) h 2 +g 3, (h) Ξ 1 ǫ +g 3 (h/ h) (1+ǫ)m/(1+m) +g 3 where B(X) c >. The second inequality holds provided h/ h 2 (so that h 2 V 1 is controlled by V ), and the last inequality holds as h provided ǫ < 1/m. The error g 3 comes from using V in the expansion of g rather than W h. We now deal with the (nearly) positive error terms g 2 and g 3. Lemma The error terms g 2 and g 3 are semi-bounded below in the following sense: if u(x) has wavefront set localized in { X ǫ(h/ h) 1/(m+1), Ξ ǫ(h/ h) m/(m+1) }, then for any δ > and N >, Op h(g j )u,u C N h (N 2)m/(m+1) δ h2m/(m+1) u 2, for j = 2,3. Proof. We prove the relevant bounds for x. The analysis for x is similar. For g 2, for N > large, and δ > small, choose < x 1 < x 2 = o(1) satisfying x 1 V (x 1 ) = h Nm/(m+1) and x 2 V (x 2 ) = h Nm/(m+1) δ. As usual, since V (x) = O(x ), the points x j, j = 1,2 satisfy x j h δ2 for any δ 2 >. The -Gevrey condition also implies x 2 x 1 h δ2 for any δ 2 > as well. To see this, Taylor s theorem says (V (x 2 ) V (x 1 )) = V (ξ)(x 2 x 1 ) for some x 1 ξ x 2. The -Gevrey condition and monotonicity near x = implies V (ξ) V (x 2 ) C V (x 2 ), so that (V (x 2 ) V (x 1 )) C V x τ 2 (x 2 ) (x 2 x 1 ) for some τ <, which in turn implies (recalling V < for x > near ) (x 2 x 1 ) C V (x 1 ) V (x ( 2 ) V (x x τ 2 = x τ 2 1 V ) (x 1 ) 2) V (x 2). x τ 2

15 RESOLVENT ESTIMATES 15 We claim V (x 1 ) V (x 2) = o(1), which will finish the proof that x 2 x 1 h δ2 for any δ 2 >. For this, we write V (x 1 ) V (x 2) = V x (x 1 ) 1 x 2 V (x x 2 2) x 1 = h δx 2 x 1. Writingx 2 = x 1 +γ(h), wearetryingtoshowγ(h) h δ2 foranyδ 2 >. Forafixed δ 2, if γ(h) h δ2 we re done. If γ(h) < h δ2, then we will produce a contradiction (in fact showing that γ(h) h δ2 ). If γ(h) < h δ2 for this δ 2, then since x 1 h δ2. Then it follows that γ(h) x 1 1, h δx 2 = h δx 1 +γ(h) 2h δ = o(1). x 1 x 1 Plugging into our earlier computation, we get x 2 x 1 = C x τ 2(1 o(1)) C δ 3 h τδ3 for any δ 3 >. Taking δ 3 > sufficiently small so that h τδ3 h δ2 implies γ(h) = x 2 x 1 h δ2, which is a contradiction to our assumption that γ(h) h δ2. Now let ψ(x) be a smooth function, ψ, ψ(x) 1 on [,x 1 ] with ψ(x) for x x 2. Assume also that k xψ C k x 2 x 1 k = o(h kδ2 ) for any δ 2 >. Let ψ(x) = ψ((h/ h) α X), α = 1/(m+1), so that We have Op h(g 2 )u,u = Op h(g 2 )(1 +2 k X ψ C k (h/ h) αk o(h kδ2 ). ψ)u,(1 ψ)u Op h(g 2 ) ψu,(1 ψ)u. + Op h(g 2 ) ψu, ψu We estimate each term separately. On the support of 1 ψ (again recalling we are only looking at x ), we have (h/ h) 1/(m+1) X x 1 so that in this region we can apply the -Gevrey condition to V 1 to absorb h 2 V 1 into V. Recall that V 1 consists of quotients of derivatives of A with powers of A. The function A is bounded above and below by a (positive) constantforxsmall, sowearereallyonlyconcernedwithestimatingafinitenumber of derivatives of A. Then according to the -Gevrey condition, for any δ 2 >, we have for some s,τ < h 2 V 1((h/ h) α X) Ch 2 x 1 sτ A ((h/ h) α X) Ch 2 sτδ2 V ((h/ h) α X),

16 16 H. CHRISTIANSON and similarly for a finite number of derivatives of V 1. By taking δ 2 > sufficiently small we see that on the support of 1 ψ, the quantization of V controls that of h 2 V 1. That is, for h > sufficiently small, Op h(g 2 )(1 ψ)u,(1 ψ)u ( h h) (1 2m) (m+1) C Op h(λ(x)v ((h/ h) α X) Ξ 1 ǫ )(1 ψ)u,(1 ψ)u. Then we calculate in this region ( h h) (1 2m)/(m+1) ( Λ(X) (h/ h) 1/(m+1) X ( ) (h/ h) 1/(m+1) XV ((h/ h) 1/(m+1) X) ) Ξ 1 ǫ = h (1 2m)/(m+1) h(2m 1)/(m+1) h Nm/(m+1) A(X,h, h) Ξ 1 ǫ where A is a symbol bounded below by a positive constant. This follows since ( ) α X x 1 h h ( ) α h h h δ2 for any δ 2 >. Taking δ 2 < α, this lower bound is (at least) a positive constant. On the set where A Ξ 1 ǫ 1, this operator is bounded below, while on the complement, we use the Sharp Gårding inequality to get for any δ 2 > Op h(g 2 )(1 ψ)u,(1 ψ)u C δ2 hh ((N 2)m+2)/(m+1) 2δ 2 h(2m 2)/(m+1) (1 ψ)u 2. Fortheremainingtwoterms, onthesupportof ψ, wehave (h/ h) 1/(m+1) X x 2. We know that xa k is an increasing function for small x, so that to estimate V 1, we estimate a finite number of derivatives of A from above, we can estimate at the right-hand endpoint x 2. That is, we have as above for s,τ < and any δ 2 >, h 2 V 1((h/ h) α X) Ch 2 x 2 sτ V (x 2 ) Ch 2 x 2 sτ 1 x 2 V (x 2 ) Ch 2 δ2(1+sτ) h Nm/(m+1) δ by our choice of x 2. This implies that on the support of ψ, h 2 V 1 is controlled by a large power of h, by taking δ 2 > sufficiently small. That is, in this region [ ( ) Λ(X) ( ) g 2 = (h/ h) (1 2m)/(m+1) (h/ h) α XV ((h/ h) α X) (h/ h) α) X ] h 2 Λ(X)V 1((h/ h) α X) Ξ 1 ǫ = h 2m/(m+1) h(2m)/(m+1) h Nm/(m+1) δ A 1 (X,h, h) Ξ 1 ǫ,

17 RESOLVENT ESTIMATES 17 where A 1 is a function satisfying k XA 1 C k,δ2 (h 1/(m+1) δ2 h 1/(m+1) ) k. Hence if δ 2 < 1/(m+1), Op h(g 2 ) ψu, ψu = O(h (N 2)m/(m+1) δ h2m/(m+1) ) u 2, and similarly Op h(g 2 ) ψu,(1 ψ)u = O(h (N 2)m/(m+1) δ h2m/(m+1) ) u 2. The proof for g 3 is the same (since we have assumed f C c G τ for some τ < ), but slightly easier, since g 3 is the error term coming from W h away from x =, and W h is already O(h 2 ). Let us recap what we have shown so far and fix some of the parameters. We have perturbed our potential by a term of size Γ, which we want to be much smaller than our lower bound on hop h (H(a)). That is, we want to solve ( ) (m 1)/(m+1) h Γ h h 1/m (h/ h) α(1+ǫ)(m 1)/m x 3 ǫ+(1+ǫ)/m h 2(m 1)/m Γ. As m will be large, ǫ < 1/m, and x = o(1), it suffices to solve or h 2m m+1 +(m 1)(1+ǫ ) m(m+1) h (m 1) (m+1) +2(m 1) m (m 1)(1+ǫ ) m(m+1) = Γ (m 1)/m, Γ = h 2m2 /(m 2 1)+(1+ǫ )/(m+1) h2 m(m 1)/(m 2 1) (1+ǫ )/(m+1). This means that for this value of Γ, our lower bound on hop h (H(a)) is Γx 3 ǫ+(1+ǫ)/m = h 2m2 /(m 2 1)+(1+ǫ )/(m+1) h2 m(m 1)/(m 2 1) (1+ǫ )/(m+1) x 3 ǫ+(1+ǫ)/m. Observe that the exponent of h is 2 + O(m 1 ) which can be made smaller than 2+η for any η > by taking m large. We also have to choose the parameter (h). For that we again match lower bounds: or (m 1)/(m+1) h3/(m+1) h (h) = Γ 1/m (h/ h) α(1+ǫ)(m 1)/m x 3 ǫ+(1+ǫ)/m h (2m 2)/m = h 2m/(m2 1)+(1+ǫ )/(m+1) (h) = h 3/(m+1) 2m/(m2 1) (1+ǫ )/(m+1) h 4 2/m (m 1)/(m2 1) (1+ǫ )/(m+1) x 3 ǫ+(1+ǫ)/m, h (m 1)/(m+1) 4+2/m+(m 1)/(m2 1)+(1+ǫ )/(m+1) x 3+ǫ (1+ǫ)/m. Taking m sufficiently large yields (h) satisfying h/ (h) = o(1), so (h) h, as required to determine x (h) and fix all the parameters.

18 18 H. CHRISTIANSON All told, we have shown for a function u(x) with semiclassical wavefront set localized in a set { X ǫ(h/ h) 1/(m+1), Ξ ǫ(h/ h) m/(m+1) } h(h/ h) (m 1)/(m+1) Op h(g)u,u CΓ(h)x 3 ǫ+(1+ǫ)/m u 2 +h(h/ h) (m 1)/(m+1) ( Op h(g 2 )u,u + Op h(g 3 )u,u ) Ch 2m2 /(m 2 1)+(1+ǫ )/(m+1) h2 m(m 1)/(m 2 1) (1+ǫ )/(m+1) x 3 ǫ+(1+ǫ)/m u 2 C N,δ h 2m/(m+1) h (N 2)m/(m+1) δ h(h/ h) (m 1)/(m+1) u 2 C h 2m2 /(m 2 1)+(1+ǫ )/(m+1) h2 m(m 1)/(m 2 1) (1+ǫ )/(m+1) x 3 ǫ+(1+ǫ)/m u 2. We note that with these parameter values, of course h/ = o(1), which implies in turn that x = o(1), and since h/ h 2, the estimate (3.11) holds, which closes the argument. This concludes the study of the principal term in the commutator expansion. Of course we still have to control the lower order terms in the commutator expansion, which we do in the following Lemma. Lemma The symbol expansion of [Q 1,a w ] in the h-weyl calculus is of the form ( (ih ) [Q 1,a w ] =Op w h 2 σ(d x,d ξ ;D y,d η ) (q 1 (x,ξ)a(y,η) q 1 (y,η)a(x,ξ)) x=y,ξ=η ) +e(x,ξ)+r 3 (x,ξ), where e satisfies Op w h(e) hop h (H(a)). Proof. Since everything is in the Weyl calculus, only the odd terms in the exponential composition expansion are non-zero. Hence the h 2 term is zero in the Weyl expansion. Now according to Lemma 2.2 and the standard L 2 continuity theorem for h-pseudodifferential operators, we need to estimate a finite number of derivatives of the error: (3.12) γ e 2 Ch 3 γ 1+γ 2=γ sup (x,ξ) T R (y,η) T R However, since q 1 (x,ξ) = ξ 2 +V,h (x), we have sup Γα,β,ρ,γ (D)(σ(D)) 3 q 1 (x,ξ)a(y,η). ρ M,ρ N 4 D x D ξ q 1 = D 3 ξq 1 =,

19 RESOLVENT ESTIMATES 19 so that σ(d) 3 q 1 (x,ξ)a(y,η) x=y,ξ=η = D 3 xq 1 D 3 ηa x=y,ξ=η = V h (x)( h/h) 3m/(m+1) Λ (( h/h) m/(m+1) η) Λ(( h/h) 1/(m+1) y)χ(y)χ(η)+r 3, where r 3 is supported in { (x,ξ) δ 1 }. Owing to the cutoffs χ(y)χ(η) in the definition of a (and the corresponding implicit cutoffs in q 1 ), we only need to estimate this error in compact sets. The derivatives h β η and h α y preserve the order of e 2 in h and increase the order in h, while the other derivatives lead to higher powers in h/ h in the symbol expansion. Hence we need only estimate e 2, as the derivatives satisfy similar estimates. In order to estimate e 2, we again use conjugation to the 2-parameter calculus, and at some point invoke the -Gevrey assumption. We have Op w h(e 2 )u = T h, hop w h(e 2 )T 1 h, h T h, h u T h, h Opw h(e 2 )T 1 h, h L 2 L 2 u, by unitarity of T h, h. But T h, hop w h (e 2)T 1 h, h = Opw h(e 2 B) and e 2 B = h 3 V h ((h/ h) 1/(m+1) X)( h/h) 3m/(m+1) Λ (Ξ) Λ(X)χ(x)χ(ξ)+r 3 B, where r 3 is again microsupported away from the critical point (coming from the derivatives on χ(x)χ(ξ). We recall that V h (x) = V (x)+w h (x), where W h(x) = Γ(h)f(x/x ). As f Cc, we know that and hence W h (x) CΓx 3, h 3 ( h/h) 3m/(m+1) Λ (Ξ)Λ(X)W h ((h/ h) 1/(m+1) X)(χ(x)χ(ξ) CΓh 3/(m+1) h3m/(m+1) x 3. As for V, since V Gτ, for x close to satisfying (in the rescaled coordinates) ( ) 1/(m+1)+ǫ1 X, ǫ 1 >, h h we have (3.13) provided h 3/(m+1) h3m/(m+1) V ((h/ h) 1/(m+1) X) ) 1/(m+1) Ch 3/(m+1) h3m/(m+1) X ( h h 2τ h 2/(m+1)+γ h3m/(m+1) V ((h/ h) 1/(m+1) X) V ((h/ h) 1/(m+1) X) 2τǫ 1 1 m+1 γ, for γ > (which of course implies we must have γ < 1/(m+1)). This can clearly be done for any τ < by taking ǫ 1 > sufficiently small.

20 2 H. CHRISTIANSON We need to estimate (3.13) in terms of V (X) Ξ 1 ǫ as (x,ξ) and (y,η) vary in (3.12). That means we need to worry about large Ξ. If Ξ δ 1 /2, say, then (3.13) is trivially bounded by If h 2/(m+1)+γ h3m/(m+1) V ((h/ h) 1/(m+1) X) Ξ 1 ǫ. Ξ max{ X 1+ǫ,δ 1 /2}, then the function g 1 c δ1, so there is nothing to prove in this region. On the other hand, if δ 1 2 Ξ X 1+ǫ, then so that Ξ 1 X 1 ǫ, ( ) α(1+ǫ) Ξ 1 ǫ X (1+ǫ)2 h h 2. Then (3.13) is bounded by ( Ch 2/(m+1)+γ h3m/(m+1) V ((h/ h) 1/(m+1) X) h h Ch 1/(m+1) hm/(m+1) V ((h/ h) 1/(m+1) X) Ξ 1 ǫ, provided (1+ǫ ) 2 ) α(1+ǫ) 2 Ξ 1 ǫ m+1 < 1 m+1 +γ, which is possible since we have already determined ǫ 1/(m + 1) and the only restriction on γ was γ < 1/(m+1). On the other hand, we have for since V (x) = O( x ), then ( ) 1/(m+1)+ǫ1 X, ǫ 1 >, h h V ((h/ h) 1/(m+1) X) = O(h ). The error term must be estimated in terms of hh(a). Recall that Λ (Ξ) C Ξ 1 ǫ, so we have shown that the error is always controlled by o(h 1/(m+1) hm/(m+1) ) Ξ 1 ǫ Λ(X)V,h((h/ h) 1/(m+1) X) +O(h ) hh(a). Finally, we are able to put things together. Let v = ϕ w u, with ϕ chosen to have support inside the set where χ(x)χ(ξ) = 1; thus the terms r and r 3 above are supported away from the support of ϕ. Then Lemma 3.12 yields i [Q 1 z,a w ]v,v = h Op w h(h(a))v,v + Op w h(e 2 )u,u CΓx 3 ǫ+(1+ǫ)/m v 2 = Ch 2m2 /(m 2 1)+3ǫ /(m 2 m) hm/(m+1) 3ǫ /(m 2 m) x 3 ǫ+(1+ǫ)/m v 2,

21 RESOLVENT ESTIMATES 21 for h sufficiently small. Here we have used the previously computed value of Γ. On the other hand, we certainly have [Q 1 z,a w ]v,v C (Q 1 z)v v, hence (Q 1 z)v CΓx 3 ǫ+(1+ǫ)/m v. We need yet compare Q to Q 1 : v CΓ 1 x 3+ǫ (1+ǫ)/m (Q 1 z)v ( CΓ 1 x 3+ǫ (1+ǫ)/m + (V,h ( Q z)v V )v ) ( ) CΓ 1 x 3+ǫ (1+ǫ)/m +Γ(h) v ( Q z)v CΓ 1 x 3+ǫ (1+ǫ)/m ( Q z)v +o(1) v provided that again ǫ is sufficiently small and m is sufficiently large. Then the term with v can be moved to the left hand side to get (now freezing h small and positive) v CΓ 1 x 3+ǫ (1+ǫ)/m ( Q z)v Ch 2m2 /(m 2 1) (1+ǫ )/(m+1) ( Q z)v = Ch 2 η ( Q z)v for η = O(m 1 ). This is (3.4). Lastly, we show how to modify the preceding argument in the case of Proposition 3.8. The main point is that the nonlinear rescaling in Γ (as part of λ) allows us to use that Γ 1/(m+1) Γ. The first step is to modify the function f and subsequently W h and V,h. Since V (x) 1 on an interval x [ a,a], with ±V (x) < for ±x > a, we choose the point x > so that xv (x) h (h), a+x x a+ǫ, and similarly for a ǫ x a x. Again we can assume that x a = o(1). Then choose f Cc (R) Gτ for some τ <, with f(x) = 1 1 2m x2m for x a+x, and f (x) or x, suppf [ a 2x,a+2x ], satisfying k xf C k x k. For our next parameter, set Γ(h) = c Γ(h) for a small constant c > to be determined, and Γ(h) the parameter computed in the case of the isolated infinitely degenerate maximum. As before then we take and let W h (x) = Γ(h)f(x), V,h (x) = V (x)+w h (x) We then follow the same arguments as in the proofs of Proposition 3.2 and 3.6, noting that the smallness assumption on the support of the microlocal cutoff ϕ in the x direction was to control lower order terms in Taylor expansions. As the

22 22 H. CHRISTIANSON function V is constant and f(x) = 1 x 2m on [ a,a], the smallness assumption translates into a small neighbourhood around [ a, a]. Hence in Proposition 3.8 we have assumed that suppϕ [ a ǫ,a + ǫ]. All of the error terms are treated similarly to the preceding proof. The only changes to check are that, since f is no longer a function of x/x, and Γ(h) = c Γ, we need to solve (in the previous notation): or h(h/ h) (m 1)/(m+1) Γ1/m (h/ h) α(1+ǫ)(m 1)/m h2(m 1)/m Γ, h(h/ h) (m 1)/(m+1) (h/ h) α(1+ǫ)(m 1)/m h2(m 1)/m (c Γ) (m 1)/m, which is true with our previous choice of Γ, provided c > is sufficiently small and independent of h. As previously, we then have (Q 1 z)v C 1 c 1/m h 2m2 /(m 2 1)+3ǫ /(m 2 m) hm/(m+1) 3ǫ /(m 2 m) v. In order to save some space, let us denote ω(h) = h 2m2 /(m 2 1)+3ǫ /(m 2 m) hm/(m+1) 3ǫ /(m 2 m). Comparing Q to Q 1 now yields: v Cc 1/m ω(h) 1 (Q 1 z)v Cc 1/m ω(h) 1( ) + (V,h ( Q z)v V )v Cc 1/m ω(h) 1( ) +Cc ( Q z)v ω(h) v Cc 1/m ω(h) 1 ( Q z)v +Cc (m 1)/m v. Freezing h > and c > sufficiently small, the term with v can be moved to the left hand side to get v Ch 2 η ( Q z)v with η = O(m 1 ) once again. This is (3.5) Infinitely degenerate and cylindrical inflection transmission trapping. In this subsection, we study the microlocal spectral theory in a neighbourhood of infinitely degenerate and cylindrical inflection transmission trapping. This is very similar to Subsection 3.4, but now the potential is assumed to be monotonic in a neighbourhood of the critical value. We begin with the case where the potential has an isolated infinitely degenerate critical point of inflection transmission type. As in the previous subsection, we write V(x) = A 2 (x) + h 2 V 1 (x) and denote V (x) = A 2 (x) to be the principal part of the potential. Let us assume the point x = 1 is an infinitely degenerate inflection point, so that locally near x = 1, the potential takes the form V (x) C 1 1 (x 1), where C 1 > 1. Of course the constant is arbitrary (chosen to again agree with those in [CM13]). Let us assume that our potential satisfies V (x) near x = 1,

23 RESOLVENT ESTIMATES 23 with V (x) < for x 1 so that the critical point x = 1 is isolated. The next Proposition says that in this case the microlocal resolvent is bounded by O(h 2 η ) for any η >. Let Q = (hd x ) 2 +V(x) z. Proposition For ǫ > sufficiently small, let ϕ S(T R) have compact support in { (x 1,ξ) ǫ}. Then for any η >, there exists C = C ǫ,η > such that (3.14) Qϕ w u C ǫ h 2+η ϕ w u, z [C 1 1 ǫ,c 1 1 +ǫ]. On the other hand, if V (x) on an interval, say x 1 [ a,a] with V (x) < for x 1 < a and x 1 > a, we do not expect anything better than Proposition The next Proposition says that this is exactly what we do get. To fix an energy level, assume V C1 1 on [ a,a]. We again write Q = (hd x ) 2 +V(x) z. Proposition For ǫ > sufficiently small, let ϕ S(T R) have compact support in { x 1 a+ǫ, ξ ǫ}. Then for any η >, there exists C = C ǫ,η > such that (3.15) Qϕ w u C ǫ h 2+η ϕ w u, z [C 1 1 ǫ,c 1 1 +ǫ]. Proof. The proof of these Propositions is again very similar, so we put them together. We will first prove Proposition 3.13, and then point out how the proof must be modified to get Proposition The idea of the proof of Proposition 3.13 (and indeed Proposition 3.14) is to round off the corners in an h-dependent fashion to obtain a finitely degenerate inflection point, and then mimic the proof of Proposition 3.4. Choose a point x = x (h) > and ǫ > such that x is the smallest number so that V (x) h (h), x x 1 ǫ, where (h) will be determined later. Similar considerations apply to choosing the parameters here as in the previous subsection, but since we wrote that in excruciating detail, we will leave out one or two details in this subsection. Fix m 2 1, and choose also an odd function f Cc ([ 2,2]) Gτ for some τ <, with f(x) = (x) 2m2+1 /(2m 2 + 1) for x 1 and f(x),f (x) for x 2. For another parameter Γ(h) to be determined, let and let and W h (x) = Γ(h)f((x 1)/x ), V,h (x) = V (x)+w h (x) V h (x) = V(x)+W h (x) (see Figure 2). The parameter Γ(h) will be seen to be a constant multiple of h 2+η, where η >, η = O(m 1 2 ) as m 2 in the case of Proposition As in the previous subsection, Γ(h) will be a small constant times this power of h in the case of Proposition By construction, V (x) V,h (x) W h Γ(h).

24 24 H. CHRISTIANSON V = C 1 x = 1 x = 1+xx = 1+2x Figure 2. The potential V and the modified potential V,h (in dashed). Let Q 1 = (hd) 2 + V h with symbol q 1 = ξ 2 + V h. The Hamilton vector field H associated to the symbol q 1 is given by H = 2ξ x V h ξ ( ) Γ(h) = 2ξ x f ((x 1)/x )+V x (x)+h 2 V 1(x) ξ. We will consider a commutant localizing in this region and singular at the critical point in a controlled way: we introduce new variables ξ x 1 Ξ = X 1 = (h/ h) β, (h/ h) α, with α,β >, α = 1/(m 2 +1), and α+β = 1 so that we may use the two-parameter calculus. We remark that in the new blown-up coordinates Ξ,X, H = (h/ h) β α( 2Ξ X (h/ h) α 2β V h((h/ h) α ) (3.16) (X 1)+1) Ξ Now fix ǫ > and set Λ 1 (s) = s s 1 ǫ ds and s Λ 2 (s) = 1+ s 1 ǫ ds. Λ 1 is a bounded symbol which looks like s near, and Λ 2 is a bounded symbol with positive derivative for s near, and Λ 2 1 everywhere. We introduce the singular symbol a(x,ξ;h) = Λ 1 (Ξ)Λ 2 (X 1)χ(x 1)χ(ξ) = Λ 1 (ξ/(h/ h) β )Λ 2 ((x 1)/(h/ h) α )χ(x 1)χ(ξ), where χ(s) is a cutoff function equal to 1 for s δ 1 and for s 2δ 1 (δ 1 will be chosen shortly). Then a is bounded since we have restricted the domain of integration to (x 1,ξ) δ 1. Further, a satisfies the symbolic estimates: X α β Ξ a C α, β.

25 RESOLVENT ESTIMATES 25 (Recall that x 1 = (h/ h) α (X 1) and ξ = (h/ h) β Ξ.) Using (3.16), it is simple to compute (3.17) with H(a) =(h/ h) β α χ(x 1)χ(ξ) ( 2Λ 1 (Ξ) X 1 1 ǫ Ξ (h/ h) α 2β V h((h/ h) α (X 1)+1) Ξ 1 ǫ Λ 2 (X 1) ) +r =:(h/ h) β α g +r suppr { x 1 > δ 1 } { ξ > δ 1 } (r comes from terms involving derivatives of χ(x 1)χ(ξ)). For X 1 (h/ h) α x we have with (h/ h) α 2β V h((h/ h) α (X 1)+1) Ξ 1 ǫ Λ 2 (X 1) = Γ(h)x (2m2+1) (h/ h) α(2m2+1) 2β (X 1) 2m2 Ξ 1 ǫ Λ 2 (X 1)+g 2, g 2 = (h/ h) α 2β (V +h 2 V 1)((h/ h) α (X 1)+1) Ξ 1 ǫ Λ 2 (X 1). Let us denote by g 1 the part of g obtained in this fashion, microlocally in { X 1 (h/ h) α x }: g 1 = g g 2 = 2Λ 1 (Ξ) X 1 1 ǫ Ξ +Γ(h)x (2m2+1) (h/ h) α(2m2+1) 2β (X 1) 2m2 Ξ 1 ǫ Λ 2 (X 1). For X 1 (h/ h) α x and Ξ (h/ h) β δ 1 consider g 1 =2Λ 1 (Ξ)Ξ X 1 1 ǫ + Γ(h) x 2m2+1 =2Λ 1 (Ξ)Ξ X 1 1 ǫ + Γ(h) x 2m2+1 (h/ h) α(2m2+1) 2β (X 1) 2m2 Λ 2 (X 1) Ξ 1 ǫ (h/ h) α (X 1) 2m2 Λ 2 (X 1) Ξ 1 ǫ, where we have used α = 1/(m 2 +1). Continuing, and rescaling using the L 2 -unitary rescaling X 1 = λ(x 1), Ξ = λ 1 Ξ, we get g 1 =λ 2( 2λ 1 Λ 1 (Ξ)(λ 1 Ξ) X 1 1 ǫ +λ 2 2m2 Γx 2m2 1 (h/ h) α Λ 2 (X 1)(λ(X 1)) 2m2 Ξ 1 ǫ ) =λ 2( 2λ 1 Λ 1 (λξ )Ξ λ 1 (X 1) 1 ǫ +λ 2 2m2 Γx 2m2 1 (h/ h) α Λ 2 (λ 1 (X 1))(X 1) 2m2 λξ 1 ǫ ). As in the previous subsection, the parameter λ > will be seen to be a small h-dependent parameter chosen to optimize lower bounds on g 1 amongst several different regions.

26 26 H. CHRISTIANSON The error term g 2 is the term in the expansion of g coming from V rather than W h. We will deal with g 2 in due course. We are now microlocalized on a set where X 1 λ(h/ h) α x, Ξ λ 1 (h/ h) β δ 1, and will be quantizing in the h-weyl calculus, so we need symbolic estimates on these sets. If X 1 λδ 1, and Ξ λ 1 δ 1, and δ 1 > is sufficiently small, then Λ 1 (λξ ) λξ and Λ 2 (λ 1 (X 1)) 1, so that g 1 is bounded below by a multiple of (3.18) min{λ 2,λ 2 2m2 Γ(h/ h) α }((Ξ ) 2 +(X 1) 2m2 ). Hence the h-quantization of g 1 is bounded below by this minimum value times h 2m2/(m2+1) on this set (using [CW11, Lemma A.2]). ( λ Now on the complementary set, if λξ max 1 (X 1) ) 1+ǫ,(δ 1 /2) 1+ǫ then g 1 cλ 2 λ 1 Λ 1 (λξ )Ξ λ 1 (X 1) 1 ǫ cλsgn(ξ )Ξ λξ 1 c for some c >. If λ 1 (X 1) 1+ǫ max ( λξ,(δ 1 /2) 1+ǫ ), we have two regions to consider. The first, if λξ (δ 1 /2) 1+ǫ (and using that λ 1 X (h/ h) α x in this region), then (3.19) (3.2) g 1 cλ 2( (Ξ ) 2 (h/ h) α(1+ǫ) x 1 ǫ ) +λ 2 2m2 Γx 2m2 1 (h/ h) α (X 1) 2m2 min{λ 2 (h/ h) α(1+ǫ) x 1 ǫ,λ 2m2 Γx 2m2 1 (h/ h) α } ((Ξ ) 2 +(X 1) 2m2 ). We optimize this by setting the two terms in the minimum equal: or λ 2 (h/ h) α(1+ǫ) x 1 ǫ = λ 2m2 Γx 2m2 1 (h/ h) α, which yields in turn the lower bound λ 2 (h/ h) α(1+ǫ) x 1 ǫ λ 2+2m2 = Γ(h/ h) αǫ x 2m2+ǫ, = Γ 1/(m2+1) (h/ h) αǫ/(m2+1)+α(1+ǫ) x ( 2m2+ǫ)/(m2+1) 1 ǫ. Then according to [CW11, Lemma A.2], the h-quantization of (3.2) is bounded below by Γ 1/(m2+1) (h/ h) αǫ/(m2+1)+α(1+ǫ) x ( 2m2+ǫ)/(m2+1) 1 ǫ h 2m2/(m2+1).